Trading Cycles for School Choice


 Elvin Harrison
 3 years ago
 Views:
Transcription
1 Trading Cycles for School Choice Marek Pycia UCLA M. Utku Ünver Boston College October 18, 2011 Abstract In this note we study the allocation and exchange of discrete resources in environments in which monetary transfers are not allowed. We allow each discrete resource to be represented by several copies, extend onto this environment the trading cycles mechanisms of Pycia and Ünver [2009], and show that the extended mechanisms are Pareto efficient and strategyproof. In particular, we construct the counterpart of Pápai [2000] hiererachical exchange mechanisms for environments with copies. Keywords: Mechanism design, group strategyproofness, Pareto efficiency, matching, house allocation, house exchange, outside options. JEL classification: C78, D78 UCLA, Department of Economics, 8283 Bunche Hall, Los Angeles, CA Boston College, Department of Economics, 140 Commonwealth Ave., Chestnut Hill, MA
2 1 Introduction The theory of mechanism design has informed the design of many markets and other institutions. Auction mechanisms have been developed to sell treasury bills, electricity, natural gas, radio spectra, timber, and foreclosed homes. Other mechanisms have been developed to allocate resources in environments in which transfers are not used or are prohibited: to allocate and exchange transplant organs such as kidneys through newly established regional programs such as the Alliance for Paired Donation (centered in Toledo, Ohio) and the New England Program for Kidney Exchange (centered in Newton, Massachusetts), and the new U.S. national program (managed by the United Network for Organ Sharing) [cf. Roth, Sönmez, and Ünver, 2004], to allocate school seats in New York City, Chicago, San Francisco, and Boston [cf. Abdulkadiroğlu and Sönmez, 2003], and to allocate dormitory rooms to students at US colleges [cf. Abdulkadiroğlu and Sönmez, 1999]. Aprimaryconcerninmarketdesignisthestrategicbehaviorofmarketparticipantsand its impact on resulting allocations. Strategic behavior becomes straightforward if the market mechanism is dominantstrategy incentive compatible: no agent can manipulate the system to his or her benefit. Nonmanipulability is not the only benefit of using dominantstrategy incentivecompatible mechanisms. Dominantstrategy incentivecompatibility imposes minimal costs of searching for and processing strategic information, and do not discriminate among agents based on their access to information and ability to strategize [cf. Vickrey, 1961, Dasgupta et al., 1979, Pathak and Sönmez, 2008]. Pycia and Ünver [2009] constructed the full class of group strategyproof and Pareto efficient mechanisms for environments without transfers in which agents have strict preferences over objects. They also show that some tradingcycles mechanisms improve upon all previously studied mechanisms, and allows one to implement efficient allocations that cannot be implemented by previously studied mechanisms. In the present note we extend their construction to environments in which each object can be represented by several copies, and show that the resultant class is strategyproof and efficient. In particular, ours is the first paper to extend Pápai [2000] hiererachical exchange mechanisms to environments with copies. Let us highlight some common features of the above market design problems with no transfers. There is a finite group of agents, each of whom would like to consume a single indivisible object to which we will refer as a house, or seat using the terminology coined by Shapley and Scarf [1974] and Abdulkadiroğlu and Sönmez, Each house has a type (for instance each seat is in a particular school). Agents have strict preferences over types and are indifferent among objects of the same type. Some of the houses might be agents 2
3 common endowment, while others belong to agents private endowments. The outcome of the problem is a matching of agents and houses. Since we provide a unified treatment of both house allocation (from social endowment) and house exchange (among agents with private endowments), we refer to our environment as house allocation and exchange. By the revelation principle, we may restrict our attention to direct revelation mechanisms; that is, agents reveal their preferences over houses, and the mechanism matches each agent with a house (or the agent s outside option). Both Pycia and Ünver [2009] and Pápai [2000] classes of mechanisms build on Gale s Top Trading Cycles (see Shapley and Scarf, 1974). Gale s mechanisms have been extended to environments with copies by Abdulkadiroğlu and Sönmez, Model 2.1 Environment Let I be a set of agents and H be a set of schools (or houses,). We use letters i, j,k to refer to agents and h, g,e to refer to school. Each school h is represented by q h copies (seats), and each agent demands one seat. Each agent i has a strict preference relation over H, denotedby i. 2 Let P i be the set of strict preference relations for agent i, andletp J denote the Cartesian product i J P i for any J I. Any profile from = ( i ) i I from P P I is called a preference profile. For all P and all J I, let J =( i ) i J P J be the restriction of to J. To simplify the exposition, we make two initial assumptions. Both of these assumptions are fully relaxed in subsequent sections. First, we initially restrict attention to allocation problems. A house allocation problem is the triple I,H, [cf. Hylland and Zeckhauser, 1979]. Throughout the paper, we fix I and H, and thus, a problem is identified with its preference profile. In Section 5, we generalize the setting and the results to house allocation and exchange by allowing agents to have initial rights over houses. The results on allocation and exchange turn out to be straightforward corollaries of the results on (pure) allocation. Second, we initially follow the tradition adopted by many papers in the literature [cf. Svensson, 1999, Bogomolnaia and Moulin, 2001] and assume that H I so that each agent is allocated a house. This assumption is satisfied in settings in which each agent is always allocated a house (there are no outside options), as well as in settings in which agents outside options are tradeable, effectively being indistinguishable from houses. In Section 6, we 1 Pycia and Ünver [2009] provide an overview of other related literature. 2 By i we denote the induced weak preference relation; that is, for any g, h H, g i h g = h or g i h. 3
4 allow for nontradeable outside options and show that analogues of our results remain true irrespective of whether H I or H < I. An outcome of a house allocation problem is a matching. To define a matching, let us start with a more general concept that we will use frequently. A submatching is an allocation of a subset of houses to a subset of agents, such that no two different agents get the same house. Formally, a submatching is a function σ : J H where J I and σ 1 (h) q h. Using the standard function notation, we denote by σ(i) the assignment of agent i J under σ, andbyσ 1 (h) the set of agents that got house h σ(j) under σ. Let S be the set of submatchings. For each σ S,letI σ denote the set of agents matched by σ. We write I σ for I I σ,andh σ for the set of houses h such that q h σ 1 (h) > 0. For all h H, lets h Sbe the set of submatchings σ Ssuch that h H σ,i.e.,the set of submatchings at which house h has unallocated copies. In virtue of the settheoretic interpretation of functions, submatchings are sets of agenthouse pairs, and are ordered by inclusion. A matching is a maximal submatching; that is, µ Sis a matching if I µ = I. Let M Sbe the set of matchings. We also write M for S M. A (direct) mechanism is a mapping ϕ : P M that assigns a matching for each preference profile (or, equivalently, allocation problem). 2.2 StrategyProofness and Pareto Efficiency Amechanismisgroupstrategyproofifthereisnogroupofagentsthatcanmisstatetheir preferences in a way such that each one in the group gets a weakly better house, and at least one agent in the group gets a strictly better house. Formally, a mechanism ϕ is group strategyproof if for all P, thereexistsnoj I and J P J such that ϕ[ J, J ](i) i ϕ[](i) for all i J, and ϕ[ J, J ](j) j ϕ[](j) for at least one j J. Amechanismϕ is (individually) strategyproof if for all P, there is no i I and i P i such that ϕ[ i, i ](i) i ϕ[](i). AmatchingisParetoefficientifnoothermatchingwouldmakeeverybodyweaklybetter off, and at least one agent strictly better off. That is, a matching µ Mis Pareto efficient if there exists no matching ν Msuch that for all i I, ν(i) i µ(i), and for some i I, ν(i) i µ(i). A mechanism is Pareto efficient if it finds a Paretoefficient matching for 4
5 every problem. 3 TradingCycles Mechanism We turn now to our extension of Pycia and Ünver [2009] trading cycles (TC). TC is a recursive algorithm that matches agents and houses in exchange cycles over a sequence of rounds. TC is more flexible, however, as it allows two types of intraround control rights over houses that agents bring to the exchange cycles: ownership and brokerage. In our description of the TTC class, each TTC mechanism was determined by a consistent ownership structure. Similarly, each TC mechanism is determined by a consistent structure of control rights. Definition 1. A structure of control rights is a collection of mappings (cσ,b σ ):H σ I σ {ownership,brokerage} σ M. The functions c σ of the control rights structure tell us which unmatched agent controls any particular unmatched house at submatching σ. Agent i controls house h H σ at submatching σ when c σ (h) =i. The type of control is determined by functions b σ.wesay that the agent c σ (h) owns h at σ if b σ (h) =ownership, and that the agent c σ (h) brokers h at σ if b σ (h) =brokerage. In the former case we call the agent an owner and the controlled house an owned house. Inthelattercaseweusethetermsbroker and brokered house. Notice that each controlled (owned or brokered) house is unmatched at σ, andanyunmatched house is controlled by some uniquely determined unmatched agent. The consistency requirement on TC control rights structures consists of three constraints on brokerage at any given submatching (the withinround requirements) and three constraints on how the control rights are related across different submatchings (the acrossrounds requirements). Withinround Requirements. Consider any σ M. (R1) There is at most one brokered house at σ. (R2) Ifi is the only unmatched agent at σ then i owns all unmatched houses at σ. (R3) Ifagenti brokers a house at σ, theni does not own any houses at σ. Furthermore, if h is brokered at σ than q h σ 1 (h) =1. 5
6 The conditions allow for different houses to be brokered at different submatchings, even though there is at most one brokered house at any given submatching. Requirements R1R2 are what we need for the TC algorithm to be well defined (R3 is necessary for Pareto efficiency and individual strategyproofness; see Appendix??). With these requirements in place, we are ready to describe the TC algorithm, postponing the introduction of the remaining consistency requirements until the next section. The TC algorithm. The algorithm consists of a finite sequence of rounds r =1, 2,... In each round some agents are matched with houses. By σ r 1 we denote the submatching of agents and houses matched before round r. Before the first round the submatching is empty, that is, σ 0 =. Ifσ r 1 M,thatis, when every agent is matched with a house, the algorithm terminates and gives matching σ r 1 as its outcome. If σ r 1 M, then the algorithm proceeds with the following three steps of round r: Step 1. Pointing. Each house h H σ r 1 points to the agent who controls it at σ r 1.Ifthereexistsabrokeratσ r 1,thenhepointstohismostpreferredhouse among the ones owned at σ r 1. Every other agent i I σ r 1 points to his most preferred house in H σ r 1. Step 2. Trading cycles. There exists n {1, 2,...} and an exchange cycle h 1 i 1 h 2...h n i n h 1 in which agents i I σ r 1 point to houses h +1 H σ r 1 agents i (here =1,..., n and superscripts are added modulo n); and houses h points to Step 3. Matching. Each agent in each trading cycle is matched with the house he is pointing to; σ r is defined as the union of σ r 1 and the set of newly matched agenthouse pairs. The algorithm terminates when all agents are matched or when no unmatched house copies remain. Looking back at the example of the previous section, we see that it was TC and that agent i 1 brokered house h 1 while other agents owned houses. We may now also see that requirements R1 and R2 are needed to ensure that in Step 1 there always is an owned house 6
7 for the broker to point to. The difference between TTC and TC is encapsulated in Step 1; the other steps are standard and were already present in Gale s TTC idea [Shapley and Scarf, 1974]. The existence of the trading cycle follows from there being a finite number of nodes (agents and houses), each pointing at another. The matching of Step 3 is well defined, as (i) each agent points to exactly one house, and (ii) each matched house is allocated to exactly one agent (no two different agents pointing to the same house h can belong to trading cycles because there is a unique pointing path that starts with house h). Finally, since we match at least one agenthouse pair in every round, and since there are finitely many agents and houses, the algorithm stops after finitely many rounds. Our algorithm builds on Gale s toptradingcycles idea, but allows more general trading cycles than top cycles. In TC, brokers do not necessarily point to their topchoice houses. In contrast, all previous developments of Gale s idea, such as the top trading cycles with newcomers [Abdulkadiroğlu and Sönmez, 1999], hierarchical exchange [Pápai, 2000], top trading cycles for school choice [Abdulkadiroğlu and Sönmez, 2003], and top trading cycles and chains [Roth, Sönmez, and Ünver, 2004], allowed only top trading cycles and had all agents point to their top choice among unmatched houses. All these previous developments may be viewed as using a subclass of TC in which all control rights are ownership rights and there are no brokers. 3 The terminology of owners and brokers is motivated by a trading analogy. In each round of the algorithm, an owner can either be matched with the house he controls or with another house obtained from an exchange. A broker cannot be matched with the house he controls; the broker can only be matched with a house obtained from an exchange with other agents. At any submatching (but not globally throughout the algorithm), we can think of the broker of house h as representing a latent agent who owns h but prefers any other house over it. The analogy is, of course, imperfect and, ultimately, our choice of terminology is arbitrary. 4 Results Introduced in the previous section, the TC algorithm with a control rights structure satisfying R1R3 provides a Paretoefficient mechanism that maps profiles from P to matchings in M. The recursive argument for the efficiency of the nonttc mechanism from Section?? applies. Proposition 1. The TC algorithm produces a Paretoefficient matching for all control rights structures that satisfy R1R3. 3 In particular, TC can easily handle private endowments, as explained in Section 5. 7
8 We are about to see that the TCinduced mapping is group strategyproof if the control rights structure also satisfies the following acrossround consistency requirements. Acrossround Requirements. Consider any submatchings σ, σ such that σ = σ +1and σ σ M, andanyagenti I σ and any house h H σ : (R4) Ifi owns house h at σ then i owns h at σ. (R5) AssumethatatleasttwoagentsfromI σ house h at σ then i brokers h at σ. own houses at σ. If i brokers (R6) Assume that at σ agent i controls h and agent i I σ controls h H σ. Then, i owns h at σ {(i, h )}, and if, in addition, i brokers h at σ but not at σ and i I σ,theni owns h at σ. Requirements R4 and R5 postulate that control rights persist: agents hold on to control rights as we move from smaller to larger submatchings, or through the rounds of the algorithm. Requirements R4 and R6 imply that whenever an agent i takes over the control of the brokered house, then the broker i is in line for ownership rights of i after i is matched (i becomes the heir to i ). We refer to this implication as a brokertoheir transition. For discussion of the assumptions see Pycia and Ünver [2009]. To sidestep the complications of R5 and R6 in the first reading, the reader is invited to keep in mind a smaller class of control rights structures in which both of these requirements are replaced by the following strong form of brokerage persistence: If σ < I 1 and agent i brokers house h at σ then i brokers h at σ. We think that by restricting attention to this smaller class of control rights structures, we are not missing much of the flexibility of the TC class of mechanism. We must stress, however, that the complication is there for a reason: see Pycia and Ünver [2009]. We are now ready to define our mechanism class and state our main results. Definition 2. Acontrolrightsstructureisconsistent if it satisfies requirements R1R6. The class of TC mechanisms (trading cycles) consists of mappings from agents preference profiles P to matchings M obtained by running the TC algorithm with consistent control rights structures. The TTC and the nonttc mechanism of Pycia and Ünver [2009] Section 3 are examples of TC. We will denote by ψ c,b the TC mechanism obtained from a consistent control rights structure {(c σ,b σ )} σ M. 8
9 Theorem 1. Every TC mechanism is individually strategyproof and Pareto efficient. The proof of this result is the same as in Pycia and Ünver [2009]. 5 House Allocation and Exchange In this section, we generalize the model by allowing agents to have private endowments. The characterizations in the resulting allocation and exchange domains are straightforward corollaries of our main results. We also relate the results to allocation and exchange market design environments. Let H = {H i } i {0} I be a collection of I +1 pairwisedisjoint subsets of H (some of which might be empty) such that i {0} I H i = H. We interpret houses from H 0 as the social endowment of the agents, and houses from H i, i I, as the private endowment of agent i. Ahouse allocation and exchange problem is a list H, I, H,. Since we allow some of the agents to have empty endowment, the allocation model of Section 2 is contained as a special case with H = {H,,..., }. We may fix H, I and H, andidentifythehouse allocation and exchange problem just by its preference profile. Matchings and mechanisms are defined as in the allocation model of Section 2. Pareto efficiency and group strategyproofness are defined in the same way as in Section 2. In particular, the equivalence between group strategyproofness and the conjunction of individual strategyproofness and nonbossiness continue to hold true. In addition to efficiency and strategyproofness, satisfactory mechanisms in this problem domain should be individually rational. A mechanism is individually rational if it always selects an individually rational matching. A matching is individually rational, if it assigns each agent a house that is at least as good as the house he would choose from among his endowment. Formally, a matching µ is individually rational if µ(i) i h i I, h H i. For agents with empty endowments, H i =, thisconditionistautologicallytrue. The following result for house allocation and exchange is now an immediate corollary of the results of previous section. Theorem 2. In house allocation and exchange problems, TC mechanisms are Pareto efficient, and strategyproof. 9
10 Furthermore, it is straightforward to identify individually rational TC mechanisms. Referring to control rights at the empty submatching as the initial control rights, let us formulate the criterion for individual rationality as follows. Proposition 2. In house allocation and exchange problems, a TC mechanism is individually rational if and only if it may be represented by a consistent control rights structure in which each agent is given the initial ownership rights of all houses from his endowment. The proof is the same as in Pycia and Ünver [2009]. The subclass of TC mechanisms without brokers is the extension of Pápai [2000] hierarchical exchange to the setting with object copies. 6 Outside Options In this final section, we drop the assumption that H I and allow agents to prefer their (nontradeable) outside options to some of the houses. Thus, some agents may be matched with their outside options, and we need to slightly modify some of the definitions. As before, I is the set of agents and H is the set of houses. Each agent i has a strict preference relation i over H and his outside option, denoted y i. We denote the set of outside options by Y. The houses preferred to the outside option are called acceptable (to the agent); the remaining houses are called unacceptable to this agent. As before, we denote by P i the set of agent i s preference profiles, and P J = i J P i for any J I. Let us initially restrict our attention to house allocation problems. This restriction can be easily relaxed as in Section 5, and we do so at the end of the section. As before, a house allocation problem is the triple I,H,. Weimposenoassumptiononthecardinalitiesof I and H; inparticular,weallowboth H I and H < I. We generalize the concept of submatching as follows: For J I, asubmatchingisa onetoone function σ : J H Y such that each agent is matched with a house or his outside option. Aterminologicalwarningisinorder. Anaturalinterpretationoftheoutsideoptionis remaining unmatched. We will not refer to the outside option in this way, however, in order to avoid confusion with our submatching terminology. As in the main body of the paper, whenever we say that an agent is unmatched at σ, werefertoagentsfromi σ = I I σ.an agent is considered matched even if he is matched to his outside option. As before, S is the set of submatchings, I σ denotes the set of agents matched by σ, H σ H denotes the set of houses with unallocated copies at σ, andweusethestandard 10
11 function notation so that σ(i) is the assignment of agent i I σ, σ 1 (h) is the set of agents that got house h σ(i σ ),andσ 1 (Y ) is the set of agents matched to their outside options. Amatchingisamaximalsubmatching,thatis,µ Sis a matching if I µ = I. As before, M Sis the set of matchings. A (direct) mechanism is a mapping ϕ : P M that assigns a matching for each preference profile (or, equivalently, allocation problem). Mechanisms, efficiency, and group strategyproofness are defined as before. The control rights structures (c, b) and their consistency R1R6 are defined as before (notice though that the meaning of some terms such as submatching has changed, as explained above). In particular, (i) only houses are owned or brokered, the outside options are not; and (ii) control rights are defined for all submatchings, including submatchings in which some agents are matched with their outside options. Notice that if a control rights structure is consistent on the domain with outside options, and H I, thentherestrictionofthe control rights structure to submatchings in which all agents are matched with houses is a consistent control rights structure in the sense of Sections 45. We will adjust the definition of the TC algorithm by adding two clauses. Clause (a). We add the following provision to Step 1 (pointing) of round r: Ifanagentprefershisoutsideoptiontoallunmatchedhouses,theagentpointstothe outside option. If there is a broker for whom the brokered house is the only acceptable house, such a broker also points to his outside option. The outside option of each agent points to the agent. Wemodifythedefinitionofσ r in Step 3 (matching); σ r is defined as the union of σ r 1 and the set of agenthouse pairs and agentoutside option pairs matched in Step 3. Clause (b). In Step 3, we do not match agents in the cycle containing the broker except if leaving this cycle unmatched implies that no cycle is matched in the current round. Clause (a) accommodates outside options. Clause (b) is added to ensure that we do not match a broker with his outside option when he prefers the brokered house to the outside option and the brokered house is not allocated to any other agent. Notice that the broker is matched only if any pointing sequence that starts with an owner ends by pointing to the broker. We will refer to the algorithm of Section 4 modified by clauses (a) and (b) as outside options TC, and when there is no risk of confusion, simply as TC. We will refer to the mechanism ψ c,b resulting from running the outside options TC on consistent control rights structures as outside options TC, ortc. Usingthesamenameisjustifiedbecausethe mechanism described above can be used to allocate houses in the setting of Sections 26, and when restricted to the case of H I and the subdomain of preferences in which 11
12 all agents prefer any house to their outside option in the setting is identical with the TC mechanism of Section 5. Indeed, in the restricted setting clause (a) is never invoked, and presence or absence of clause (b) has no impact on the allocation. This follows from the group strategyproofness of TC of Section 5. Given a profile of agents preferences, agents who are brokers along the run of the TC without clause (b) can replicate the run of TC with clause (b) as follows: The first agent who becomes a broker along the path of the algorithm reports all houses that are matched in cycles not involving the broker ahead of the house the broker will be allocated, while keeping his preference profile otherwise intact. If another agent becomes a broker after the first broker is matched or loses his brokerage right, we modify this agent s preferences in the same way, and the same for other brokers. If the outcomes of the mechanism were dependent on whether the brokers simulate clause (b) or not, there would be a preference profile in which one of the brokers could either improve his outcome or boss other agents; contrary to group strategyproofness. By Theorem 1 this is not possible. The fact that clause (b) does not impact allocation in the setting without outside options is analogous to the wellknown fact that in TTC the order in which we match the cycles of agents does not matter. Theorem 3. In the environment with outside options, every TC mechanism is strategyproof and Pareto efficient. We are now ready to extend the characterization to the general allocation and exchange setting with outside options. As in Section 5, the social endowment H 0 H and agents endowments H i H, i I, aredisjointandsumuptoh. Ahouseallocationandexchange problem is a list H, I, H, where H = {H i } i {0} I. The results of Section 5 translate to the setting with outside options. Theorem 4. In house allocation and exchange problems with outside options, TC mechanism are Pareto efficient, and strategyproof. As before, it is straightforward to identify individually rational TC mechanisms. Proposition 3. In house allocation and exchange problems with outside options, a TC mechanism is individually rational if and only if it may be represented by a consistent control rights structure in which each agent is given the initial ownership rights of all houses from his endowment. In the environment with outside options, we define the TTC mechanisms as TC with no brokers. 12
13 References Atila Abdulkadiroğlu and Tayfun Sönmez. House allocation with existing tenants. Journal of Economic Theory, 88: ,1999. Atila Abdulkadiroğlu and Tayfun Sönmez. School choice: A mechanism design approach. American Economic Review, 93: ,2003. Anna Bogomolnaia and Herve Moulin. A new solution to the random assignment problem. Journal of Economic Theory, 100: ,2001. Partha Dasgupta, Peter Hammond, and Eric Maskin. The implementation of social choice rules: Some general results on incentive compatibility. Review of Economic Studies, 46: , Aanund Hylland and Richard Zeckhauser. The efficient allocation of individuals to positions. Journal of Political Economy, 87: ,1979. Szilvia Pápai. Strategyproof assignment by hierarchical exchange. Econometrica, 68: , Parag A. Pathak and Tayfun Sönmez. Leveling the playing field: Sincere and sophisticated players in the boston mechanism. American Economic Review, 98: , Marek Pycia and M. Utku Ünver. Incentive compatible allocation and exchange of discrete resources. UCLA and Boston College, unpublished mimeo, Alvin E. Roth, Tayfun Sönmez, and M. Utku Ünver. Kidney exchange. Quarterly Journal of Economics, 119: ,2004. Lloyd Shapley and Herbert Scarf. On cores and indivisibility. Journal of Mathematical Economics, 1:23 37,1974. LarsGunnar Svensson. Strategyproof allocation of indivisible goods. Social Choice and Welfare, 16: ,1999. William Vickrey. Counterspeculation, auctions and competitive sealed tenders. Journal of Finance, 16:8 37,
The Role of Priorities in Assigning Indivisible Objects: A Characterization of Top Trading Cycles
The Role of Priorities in Assigning Indivisible Objects: A Characterization of Top Trading Cycles Atila Abdulkadiroglu and Yeonoo Che Duke University and Columbia University This version: November 2010
More informationHousing Markets & Top Trading Cycles. Tayfun Sönmez. 16th Jerusalem Summer School in Economic Theory Matching, Auctions, and Market Design
Housing Markets & Top Trading Cycles Tayfun Sönmez 16th Jerusalem Summer School in Economic Theory Matching, Auctions, and Market Design House Allocation Problems: A Collective Ownership Economy A house
More informationTop Trading with Fixed TieBreaking in Markets with Indivisible Goods
Top Trading with Fixed TieBreaking in Markets with Indivisible Goods Lars Ehlers March 2012 Abstract We study markets with indivisible goods where monetary compensations are not possible. Each individual
More informationAn Alternative Characterization of Top Trading Cycles
An Alternative Characterization of Top Trading Cycles Thayer Morrill December 2011 Abstract This paper introduces two new characterizations of the Top Trading Cycles algorithm. The key to our characterizations
More informationTwo Simple Variations of Top Trading Cycles
Economic Theory manuscript No. (will be inserted by the editor) Two Simple Variations of Top Trading Cycles Thayer Morrill May 2014 Abstract Top Trading Cycles is widely regarded as the preferred method
More informationMaking Efficient School Assignment Fairer
Making Efficient School Assignment Fairer Thayer Morrill November, 2013 Abstract Surprisingly, nearly every school district that utilizes a centralized assignment procedure makes a Pareto inefficient assignment.
More informationMatching Markets: Theory and Practice
Matching Markets: Theory and Practice Atila Abdulkadiroğlu Duke University Tayfun Sönmez Boston College 1 Introduction It has been almost half a century since David Gale and Lloyd Shapley published their
More informationWhat s the Matter with Tiebreaking? Improving Efficiency in School Choice
What s the Matter with Tiebreaking? Improving Efficiency in School Choice By Aytek Erdil and Haluk Ergin Abstract In several school choice districts in the United States, the student proposing deferred
More informationTwoSided Matching Theory
TwoSided Matching Theory M. Utku Ünver Boston College Introduction to the Theory of TwoSided Matching To see which results are robust, we will look at some increasingly general models. Even before we
More information17.6.1 Introduction to Auction Design
CS787: Advanced Algorithms Topic: Sponsored Search Auction Design Presenter(s): Nilay, Srikrishna, Taedong 17.6.1 Introduction to Auction Design The Internet, which started of as a research project in
More informationTiers, Preference Similarity, and the Limits on Stable Partners
Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider
More informationThe Equitable Top Trading Cycles Mechanism for School Choice
Rustamdjan Hakimov Onur Kesten The Equitable Top Trading Cycles Mechanism for School Choice Discussion Paper SP II 2014 210 November 2014 (WZB) Berlin Social Science Center Research Area Markets and Choice
More informationAge Based Preferences in Paired Kidney Exchange
Age Based Preferences in Paired Kidney Exchange Antonio Nicolò Carmelo RodríguezÁlvarez November 1, 2013 Abstract We consider a model of Paired Kidney Exchange (PKE) with feasibility constraints on the
More informationInvestigación Operativa. The uniform rule in the division problem
Boletín de Estadística e Investigación Operativa Vol. 27, No. 2, Junio 2011, pp. 102112 Investigación Operativa The uniform rule in the division problem Gustavo Bergantiños Cid Dept. de Estadística e
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationThe Design of School Choice Systems in NYC and Boston: GameTheoretic Issues
The Design of School Choice Systems in NYC and Boston: GameTheoretic Issues Alvin E. Roth joint work with Atila Abdulkadiroğlu, Parag Pathak, Tayfun Sönmez 1 Outline of today s class NYC Schools: design
More informationWorking Paper Secure implementation in ShapleyScarf housing markets
econstor www.econstor.eu Der OpenAccessPublikationsserver der ZBW LeibnizInformationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Fujinaka,
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationSchool Admissions Reform in Chicago and England: Comparing Mechanisms by their Vulnerability to Manipulation
School Admissions Reform in Chicago and England: Comparing Mechanisms by their Vulnerability to Manipulation Parag A. Pathak Tayfun Sönmez This version: January 2011 Abstract In Fall 2009, officials from
More informationProperties of Stabilizing Computations
Theory and Applications of Mathematics & Computer Science 5 (1) (2015) 71 93 Properties of Stabilizing Computations Mark Burgin a a University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA
More informationWorking Paper Series ISSN 14240459. Working Paper No. 238. Stability for Coalition Formation Games. Salvador Barberà and Anke Gerber
Institute for Empirical Research in Economics University of Zurich Working Paper Series ISSN 14240459 Working Paper No. 238 A Note on the Impossibility of a Satisfactory Concept of Stability for Coalition
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationEffective affirmative action in school choice
Theoretical Economics 8 (2013), 325 363 15557561/20130325 Effective affirmative action in school choice Isa E. Hafalir Tepper School of Business, Carnegie Mellon University M. Bumin Yenmez Tepper School
More informationLecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationeach college c i C has a capacity q i  the maximum number of students it will admit
n colleges in a set C, m applicants in a set A, where m is much larger than n. each college c i C has a capacity q i  the maximum number of students it will admit each college c i has a strict order i
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationGains from Trade. Christopher P. Chambers and Takashi Hayashi. March 25, 2013. Abstract
Gains from Trade Christopher P. Chambers Takashi Hayashi March 25, 2013 Abstract In a market design context, we ask whether there exists a system of transfers regulations whereby gains from trade can always
More informationEconomics of Insurance
Economics of Insurance In this last lecture, we cover most topics of Economics of Information within a single application. Through this, you will see how the differential informational assumptions allow
More information5 Directed acyclic graphs
5 Directed acyclic graphs (5.1) Introduction In many statistical studies we have prior knowledge about a temporal or causal ordering of the variables. In this chapter we will use directed graphs to incorporate
More informationBasic Probability Concepts
page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationThis asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationHybrid Auctions Revisited
Hybrid Auctions Revisited Dan Levin and Lixin Ye, Abstract We examine hybrid auctions with affiliated private values and riskaverse bidders, and show that the optimal hybrid auction trades off the benefit
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitelyrepeated prisoner s dilemma
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationMatching with Waiting Times: The German EntryLevel Labor Market for Lawyers
Matching with Waiting Times: The German EntryLevel Labor Market for Lawyers Philipp D. Dimakopoulos and C.Philipp Heller Humboldt University Berlin 8th Matching in Practice Workshop, Lisbon, 2014 Dimakopoulos
More informationRegular Languages and Finite Automata
Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a
More informationIncentives in Landing Slot Problems
Incentives in Landing Slot Problems James Schummer Azar Abizada February 12, 2013 Abstract We analyze three applied incentive compatibility conditions within a class of queueing problems motivated by the
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationLecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
More informationChristmas Gift Exchange Games
Christmas Gift Exchange Games Arpita Ghosh 1 and Mohammad Mahdian 1 Yahoo! Research Santa Clara, CA, USA. Email: arpita@yahooinc.com, mahdian@alum.mit.edu Abstract. The Christmas gift exchange is a popular
More informationLecture 22: November 10
CS271 Randomness & Computation Fall 2011 Lecture 22: November 10 Lecturer: Alistair Sinclair Based on scribe notes by Rafael Frongillo Disclaimer: These notes have not been subjected to the usual scrutiny
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The CoxRossRubinstein
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationReading 13 : Finite State Automata and Regular Expressions
CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model
More informationMultiunit auctions with budgetconstrained bidders
Multiunit auctions with budgetconstrained bidders Christian Borgs Jennifer Chayes Nicole Immorlica Mohammad Mahdian Amin Saberi Abstract We study a multiunit auction with multiple agents, each of whom
More informationStable matching: Theory, evidence, and practical design
THE PRIZE IN ECONOMIC SCIENCES 2012 INFORMATION FOR THE PUBLIC Stable matching: Theory, evidence, and practical design This year s Prize to Lloyd Shapley and Alvin Roth extends from abstract theory developed
More informationLecture 2: Universality
CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal
More informationSettling a Question about Pythagorean Triples
Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands EMail address:
More informationOn the Existence of Nash Equilibrium in General Imperfectly Competitive Insurance Markets with Asymmetric Information
analysing existence in general insurance environments that go beyond the canonical insurance paradigm. More recently, theoretical and empirical work has attempted to identify selection in insurance markets
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationIt all depends on independence
Working Papers Institute of Mathematical Economics 412 January 2009 It all depends on independence Daniel Eckert and Frederik Herzberg IMW Bielefeld University Postfach 100131 33501 Bielefeld Germany email:
More informationPreferences. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Preferences 1 / 20
Preferences M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Preferences 1 / 20 Preference Relations Given any two consumption bundles x = (x 1, x 2 ) and y = (y 1, y 2 ), the
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationGames Manipulators Play
Games Manipulators Play Umberto Grandi Department of Mathematics University of Padova 23 January 2014 [Joint work with Edith Elkind, Francesca Rossi and Arkadii Slinko] GibbardSatterthwaite Theorem All
More informationRegular Expressions with Nested Levels of Back Referencing Form a Hierarchy
Regular Expressions with Nested Levels of Back Referencing Form a Hierarchy Kim S. Larsen Odense University Abstract For many years, regular expressions with back referencing have been used in a variety
More information(a) Write each of p and q as a polynomial in x with coefficients in Z[y, z]. deg(p) = 7 deg(q) = 9
Homework #01, due 1/20/10 = 9.1.2, 9.1.4, 9.1.6, 9.1.8, 9.2.3 Additional problems for study: 9.1.1, 9.1.3, 9.1.5, 9.1.13, 9.2.1, 9.2.2, 9.2.4, 9.2.5, 9.2.6, 9.3.2, 9.3.3 9.1.1 (This problem was not assigned
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationA Simple Characterization for TruthRevealing SingleItem Auctions
A Simple Characterization for TruthRevealing SingleItem Auctions Kamal Jain 1, Aranyak Mehta 2, Kunal Talwar 3, and Vijay Vazirani 2 1 Microsoft Research, Redmond, WA 2 College of Computing, Georgia
More informationSchooling, Political Participation, and the Economy. (Online Supplementary Appendix: Not for Publication)
Schooling, Political Participation, and the Economy Online Supplementary Appendix: Not for Publication) Filipe R. Campante Davin Chor July 200 Abstract In this online appendix, we present the proofs for
More informationSocial Networks and Stable Matchings in the Job Market
Social Networks and Stable Matchings in the Job Market Esteban Arcaute 1 and Sergei Vassilvitskii 2 1 Institute for Computational and Mathematical Engineering, Stanford University. arcaute@stanford.edu
More informationMarkov random fields and Gibbs measures
Chapter Markov random fields and Gibbs measures 1. Conditional independence Suppose X i is a random element of (X i, B i ), for i = 1, 2, 3, with all X i defined on the same probability space (.F, P).
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationAssignment of Arrival Slots
Assignment of Arrival Slots James Schummer Rakesh V. Vohra June 1, 2012 Abstract Industry participants agree that, when inclement weather forces the FAA to reassign airport landing slots, incentives and
More informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationA Simple Model of Price Dispersion *
Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion
More informationRegular Expressions and Automata using Haskell
Regular Expressions and Automata using Haskell Simon Thompson Computing Laboratory University of Kent at Canterbury January 2000 Contents 1 Introduction 2 2 Regular Expressions 2 3 Matching regular expressions
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationCS 3719 (Theory of Computation and Algorithms) Lecture 4
CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 ChurchTuring thesis Let s recap how it all started. In 1990, Hilbert stated a
More informationOnline Appendix. A2 : Description of New York City High School Admissions
Online Appendix This appendix provides supplementary material for Strategyproofness versus Efficiency in Matching with Indifferences: Redesigning the NYC High School Match. The numbering of sections parallels
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More informationWorking Paper Secure implementation in economies with indivisible objects and money
econstor www.econstor.eu Der OpenAccessPublikationsserver der ZBW LeibnizInformationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Fujinaka,
More informationMatching with (BranchofChoice) Contracts at United States Military Academy
Matching with (BranchofChoice) Contracts at United States Military Academy Tayfun Sönmez and Tobias B. Switzer May 2011 Abstract Branch selection is a key decision in a cadet s military career. Cadets
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationTHE ROOMMATES PROBLEM DISCUSSED
THE ROOMMATES PROBLEM DISCUSSED NATHAN SCHULZ Abstract. The stable roommates problem as originally posed by Gale and Shapley [1] in 1962 involves a single set of even cardinality 2n, each member of which
More informationManipulability of the Price Mechanism for Data Centers
Manipulability of the Price Mechanism for Data Centers Greg Bodwin 1, Eric Friedman 2,3,4, and Scott Shenker 3,4 1 Department of Computer Science, Tufts University, Medford, Massachusetts 02155 2 School
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More informationSharing Online Advertising Revenue with Consumers
Sharing Online Advertising Revenue with Consumers Yiling Chen 2,, Arpita Ghosh 1, Preston McAfee 1, and David Pennock 1 1 Yahoo! Research. Email: arpita, mcafee, pennockd@yahooinc.com 2 Harvard University.
More informationGalois Theory III. 3.1. Splitting fields.
Galois Theory III. 3.1. Splitting fields. We know how to construct a field extension L of a given field K where a given irreducible polynomial P (X) K[X] has a root. We need a field extension of K where
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More information4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationWorking Paper Series
RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos HervésBeloso, Emma Moreno García and
More informationMINIMAL BOOKS OF RATIONALES
MINIMAL BOOKS OF RATIONALES José Apesteguía Miguel A. Ballester D.T.2005/01 MINIMAL BOOKS OF RATIONALES JOSE APESTEGUIA AND MIGUEL A. BALLESTER Abstract. Kalai, Rubinstein, and Spiegler (2002) propose
More informationVickreyDutch Procurement Auction for Multiple Items
VickreyDutch Procurement Auction for Multiple Items Debasis Mishra Dharmaraj Veeramani First version: August 2004, This version: March 2006 Abstract We consider a setting where there is a manufacturer
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationBrewer s Conjecture and the Feasibility of Consistent, Available, PartitionTolerant Web Services
Brewer s Conjecture and the Feasibility of Consistent, Available, PartitionTolerant Web Services Seth Gilbert Nancy Lynch Abstract When designing distributed web services, there are three properties that
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More information